Frequency and mass optimization for an axially functionally graded GNP-reinforced conical shell with variable thickness

doi:10.1007/s10483-026-3373-7

Abstract

Abstract:

This paper studies the optimization of mass and frequency for a polymeric conical shell reinforced with graphene nanoplatelets (GNPs). The volume fraction of the GNPs and the thickness of the shell change along the meridional direction. The modeling of the conical shell is conducted by the first-order shear deformation theory (FSDT), and the governing equations and boundary conditions are derived by Hamilton’s principle. A semi-analytical solution is presented, including an analytical solution carried out in the circumferential direction and a numerical solution conducted in the meridional direction utilizing the differential quadrature method (DQM). To maximize the fundamental frequency and minimize the mass, the particle swarm optimization (PSO) is utilized, taking into account some constraints on the minimum thickness of the shell and the maximum volume fraction of the GNPs. The optimization process involves finding the optimal profiles of thickness and volume fraction of the GNPs.

Key words: optimization, free vibration, conical shell, graphene nanoplatelet (GNP), variable thickness

2010 MSC Number:

TB332

Zhigang ZHAO, Jun GAO, Feng LI, H. AFSHARI. Frequency and mass optimization for an axially functionally graded GNP-reinforced conical shell with variable thickness. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 859-882.

TrendMD

Figures/Tables 15

Fig. 1

Fig. 2

Table 1

Convergence analysis of the numerical solution presented in the meridional direction"

Condition		$N = 5$	$N = 7$	$N = 9$	$N = 11$	$N = 13$	$N = 15$	$N = 17$	$N = 19$
CC	$λ 31$	0.282 0	0.277 9	0.276 7	0.276 6	0.276 7	0.276 7	0.276 7	0.276 7
	$λ 32$	1.777 0	0.517 8	0.517 4	0.517 1	0.517 2	0.517 3	0.517 3	0.517 3
	$λ 33$	2.530 2	0.685 1	0.700 4	0.699 9	0.699 8	0.699 9	0.699 9	0.699 9
	$λ 34$	2.813 9	2.524 4	0.807 0	0.805 1	0.803 1	0.803 1	0.803 1	0.803 1
SC	$λ 31$	0.267 1	0.260 6	0.259 5	0.259 4	0.259 4	0.259 3	0.259 3	0.259 3
	$λ 32$	1.543 7	0.507 7	0.507 0	0.506 9	0.506 9	0.506 9	0.506 9	0.506 9
	$λ 33$	2.484 1	0.698 0	0.694 3	0.694 5	0.694 4	0.694 4	0.694 4	0.694 4
	$λ 34$	2.903 7	2.486 4	0.801 5	0.801 4	0.799 5	0.799 5	0.799 5	0.799 5
CS	$λ 31$	0.214 9	0.222 3	0.222 5	0.222 3	0.222 1	0.222 0	0.221 9	0.221 9
	$λ 32$	1.597 5	0.516 4	0.516 3	0.515 7	0.515 7	0.515 8	0.515 8	0.515 8
	$λ 33$	1.824 5	0.677 6	0.690 5	0.690 5	0.690 8	0.690 9	0.690 9	0.690 9
	$λ 34$	2.764 1	1.825 3	0.792 2	0.786 4	0.786 5	0.786 4	0.786 3	0.786 3
SS	$λ 31$	0.204 1	0.202 5	0.202 4	0.202 4	0.202 4	0.202 4	0.202 3	0.202 3
	$λ 32$	1.327 7	0.505 4	0.505 4	0.505 3	0.505 2	0.505 2	0.505 2	0.505 2
	$λ 33$	1.818 2	0.689 3	0.685 6	0.685 5	0.685 5	0.685 5	0.685 5	0.685 5
	$λ 34$	2.756 2	1.821 1	0.785 7	0.782 9	0.783 1	0.783 1	0.783 1	0.783 1
FC	$λ 31$	0.133 9	0.149 0	0.152 1	0.152 7	0.152 9	0.152 9	0.152 9	0.152 9
	$λ 32$	0.353 4	0.355 0	0.359 0	0.360 1	0.360 5	0.360 5	0.360 5	0.360 5
	$λ 33$	2.387 3	0.594 8	0.598 1	0.598 8	0.599 1	0.599 2	0.599 2	0.599 2
	$λ 34$	2.479 8	0.743 4	0.746 9	0.746 1	0.745 8	0.745 8	0.745 8	0.745 8
CF	$λ 31$	0.076 0	0.059 1	0.051 5	0.047 1	0.044 8	0.044 0	0.044 0	0.044 1
	$λ 32$	0.237 6	0.268 0	0.277 6	0.281 1	0.282 3	0.282 4	0.282 2	0.282 1
	$λ 33$	1.789 0	0.594 7	0.592 0	0.589 6	0.588 9	0.588 8	0.588 9	0.589 0
	$λ 34$	1.979 7	0.708 7	0.731 6	0.733 4	0.734 3	0.734 2	0.734 1	0.734 0

Table 1

Fig. 3

Table 2

Dimensionless natural frequencies of an SC homogeneous isotropic conical shell with variable thickness"

Frequency	$l = 1$	$l = 2$	$l = 3$	$l = 4$	$l = 5$	$l = 6$
Present	0.039 8	0.061 2	0.075 3	0.090 8	0.107 9	0.127 0
Amirabadi et al.^[43]	0.039 8	0.061 2	0.075 3	0.090 8	0.107 9	0.127 0

Table 2

Table 3

Natural frequencies (kHz) of a CS GNP-reinforced conical shell of uniform thickness and uniformly distributed GNPs"

Frequency	$n = 1$				$n = 2$
Frequency	$l = 1$	$l = 2$	$l = 3$	$l = 4$	$l = 1$	$l = 2$	$l = 3$	$l = 4$
Present	0.310 5	0.389 2	0.497 1	0.565 7	0.180 7	0.369 4	0.497 4	0.624 7
Afshari^[2]	0.310 5	0.389 2	0.497 1	0.565 7	0.180 7	0.369 4	0.497 5	0.624 8
Frequency	$n = 3$				$n = 4$
Frequency	$l = 1$	$l = 2$	$l = 3$	$l = 4$	$l = 1$	$l = 2$	$l = 3$	$l = 4$
Present	0.151 8	0.313 7	0.458 9	0.608 1	0.190 1	0.326 4	0.469 7	0.628 3
Afshari^[2]	0.151 8	0.313 7	0.458 9	0.608 2	0.190 1	0.326 4	0.469 9	0.628 5

Table 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Table 4

Optimum design of the shell under various boundary conditions"

Parameter	CC	SC	CS	SS	FC	CF
$n f$	6	5	6	5	3	2
α	$− 0.474 8$	$− 0.540 3$	$− 0.463 3$	$− 0.456 2$	$− 0.458 0$	0.969 7
β	3.387 3	2.769 5	3.512 5	3.592 8	3.572 7	3.376 9
p	1.288 6	1.681 9	1.526 1	1.636 7	1.359 8^*	1.006 5
q	1.455 4	0.879 2	0.772 4	0.323 7	0.000 0	2.311 7
$Ω f / (rad ⋅ s − 1)$	971.23	940.57	845.29	822.33	824.73	474.91
λ	0.179 7	0.174 0	0.156 4	0.152 1	0.152 6	0.087 9
$m s / kg$	15.15	15.22	15.14	15.14	15.14	21.05
$F (= m s / Ω f) / N$	0.015 6	0.016 2	0.017 9	0.018 4	0.018 4	0.044 3
*Since $q = 0,$ the value of p is irrelevant to the FC shell

Table 4

Fig. 10

Fig. 11

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